Electrical Engineering 2021 Paper I 50 marks Calculate

Q8

For an FM communication system with β = 2 and white channel noise with PSD Sₙ(ω) = 10⁻¹⁰, the output SNR is found to be 28 dB. The base band signal m(t) is Gaussian, band-limited to 15 kHz, and 3σ loading is used. Determine the following : (i) The received signal power 'Sᵢ' (ii) The output signal power 'S₀' (iii) The output noise power 'N₀' (20 marks) (b) A three-phase, 4-pole, 50 Hz induction motor has a rotor resistance of 4·5 Ω/phase and a standstill reactance of 8·5 Ω/phase with no external resistance in the rotor circuit. The starting torque of the motor is 85 Nm. Neglecting stator voltage drop, determine the following : (i) The rotor voltage at standstill (ii) The starting torque, if a 3 Ω resistance were added in each rotor phase (iii) The rotor induced voltage and the torque at a slip of 0·03 (20 marks) (c) A 220 V, 1500 rpm, 10 A separately excited dc motor has an armature resistance of 1 ohm. It is fed from a single phase fully-controlled bridge rectifier with an ac source voltage of 230 V, 50 Hz. Assuming continuous load current, determine the following : (i) Motor speed at the firing angle of 30° and torque of 5 Nm (ii) Developed torque at the firing angle of 45° and speed of 1000 rpm (10 marks)

हिंदी में प्रश्न पढ़ें

β = 2 तथा श्वेत चैनल रव जिसमें PSD Sₙ(ω) = 10⁻¹⁰ है, से संयुक्त एक FM संचार प्रणाली के लिए निर्गत SNR = 28 dB पाया गया है । आधार पटिका (बेस बैंड) संकेत m(t) गॉसियन है जो 15 kHz पर पटिका-सीमित है, तथा 3σ भारण (लोडिंग) का प्रयोग किया गया है । निम्नलिखित का मान ज्ञात कीजिए : (i) प्राप्त संकेत शक्ति 'Sᵢ' (ii) निर्गत संकेत शक्ति 'S₀' (iii) निर्गत रव शक्ति 'N₀' (20 अंक) (b) एक त्रि-कला, 4-ध्रुव, 50 Hz प्रेरण मोटर का घूर्णक प्रतिरोध 4·5 Ω/कला तथा गतिहीन अवस्था में, घूर्णक परिपथ में किसी बाह्य प्रतिरोध के बिना प्रतिघात 8·5 Ω/कला है । मोटर का प्रवर्तन बल-आघूर्ण 85 Nm है । स्टेटर के वोल्टता पात (गिरावट) को नगण्य मानते हुए, निम्नलिखित का मान ज्ञात कीजिए : (i) गतिहीन अवस्था में घूर्णक की वोल्टता (ii) यदि घूर्णक की प्रत्येक कला में 3 Ω प्रतिरोध जोड़ा गया हो, तो प्रवर्तन बल-आघूर्ण (iii) 0·03 सर्पण (स्लिप) की अवस्था में घूर्णक की प्रेरित वोल्टता तथा बल-आघूर्ण (20 अंक) (c) एक 220 V, 1500 rpm, 10 A, अलग (पृथक) से उत्तेजित dc मोटर का आर्मेचर प्रतिरोध 1 ओम है । इसे एक 230 V के ac स्रोत वोल्टता, 50 Hz से संयोजित एक कला पूर्ण-नियंत्रित सेतु दिष्टकारी से भारित किया जाता है । भार धारा को सतत मानते हुए निम्नलिखित का मान ज्ञात कीजिए : (i) 30° प्रसर्जन कोण तथा 5 Nm बल-आघूर्ण पर मोटर की गति (ii) 45° प्रसर्जन कोण तथा 1000 rpm गति पर विकसित बल-आघूर्ण (10 अंक)

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How this answer will be evaluated

Approach

Calculate the required parameters for all three parts systematically. Spend approximately 40% of time on part (a) FM communication (20 marks), 40% on part (b) induction motor (20 marks), and 20% on part (c) DC motor control (10 marks). Begin with stating relevant formulas, substitute given values with proper units, perform step-by-step calculations, and conclude with final numerical answers in appropriate units (W, dB, V, Nm, rpm).

Key points expected

  • Part (a): Apply FM SNR formula (S₀/N₀) = 3β²(Δf/fm)²(Si/Ni) with β=2, 3σ loading factor, and convert 28 dB SNR to linear scale; calculate Si using noise power Ni = η×B
  • Part (a): Compute output signal power S₀ using S₀ = (kf²/2π)×Pm and output noise power N₀ from total SNR relationship
  • Part (b): Calculate rotor standstill voltage E₂ using starting torque formula Tst = (3/ωs)×(E₂²R₂)/(R₂²+X₂²) with given 85 Nm torque
  • Part (b): Determine modified starting torque with added 3Ω resistance using Tst ∝ R₂/(R₂²+X₂²) and find slip=0.03 values using torque-slip characteristics
  • Part (c): Compute motor back EMF constant Kφ from rated conditions, then find speed at α=30° using Vt = (2√2×230/π)cosα = Ea + IaRa with T=5 Nm
  • Part (c): Calculate developed torque at α=45° and 1000 rpm using rectifier output voltage equation and torque-speed relationship

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies and applies: for (a) Carson's rule, FM threshold, and SNR improvement with 3σ loading; for (b) torque-slip characteristics and Thevenin equivalent of rotor circuit; for (c) separately excited DC motor equations and rectifier average voltage with firing angle controlUses correct basic formulas but makes minor errors in applying 3σ loading concept or confuses standstill/reactance values; understands torque-slip relationship but applies approximate methodsConfuses FM with AM noise analysis, uses incorrect torque formula (omits 3/ωs factor), or applies DC shunt motor equations instead of separately excited configuration
Numerical accuracy25%12.5All calculations precise with correct unit conversions (dB to linear, rpm to rad/s), proper handling of √2 factors in rectifier equations, and final answers with appropriate significant figures (Si ≈ 0.5-1 μW range, S₀ in mW, rotor voltage ~100-150V)Correct methodology but arithmetic errors in final multiplication or unit conversion mistakes (e.g., kHz vs Hz, degrees vs radians for firing angle)Order of magnitude errors in power calculations, incorrect slip values, or fundamental mistakes like using line voltage instead of phase voltage in torque calculations
Diagram quality10%5Includes clear phasor diagram for rotor impedance at standstill in (b), torque-slip characteristic curve showing original and modified resistance cases, and schematic of fully-controlled bridge rectifier feeding DC motor in (c) with firing angle markedBasic sketches present but missing key labels (slip axis, synchronous speed point, or thyristor numbering in bridge)No diagrams despite need for visual representation of phase relationships or circuit topology; or completely incorrect circuit diagrams
Step-by-step derivation25%12.5Shows complete derivation chain: for (a) from SNR definition → β relation → Si calculation → S₀/N₀ split; for (b) explicit torque equation derivation with substitution; for (c) rectifier output voltage derivation integrating over conduction periodStates final formulas with direct substitution but skips intermediate algebraic steps; jumps from given data to answer without showing manipulationOnly writes final answers with no derivation; or incorrect formula substitution without explaining variable meanings
Practical interpretation20%10Interprets results practically: for (a) comments on FM threshold and why 3σ loading matters for broadcast quality; for (b) discusses why external rotor resistance improves starting torque but reduces running efficiency; for (c) relates firing angle control to industrial speed regulation applications like textile millsBrief mention of practical significance without elaboration; generic statements about 'motor control is important'No physical interpretation of numerical results; fails to recognize that negative firing angle or impossible speed values indicate calculation errors

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